Abstract

In this article, a new concept of neutrosophic soft quad-topological structures is introduced. Generalised neutrosophic soft open sets in neutrosophic soft quad-topological structures concerning soft points of the space are introduced. Different results are addressed in neutrosophic soft quad-topological structure on the basis of these new neutrosophic soft open sets. Neutrosophic soft separation axioms and other separation axioms are addressed in neutrosophic soft quad-topological structures. The engagement of these axioms is switched over with different results with respect to soft points. Neutrosophic soft topological properties of some results are also addressed in neutrosophic soft quad-topological structure. To secure the results, examples are constructed. The nonvalidity of some results are justified with examples. The understanding of some complicated problems is secured with simple techniques.

1. Introduction

Zadeh [1] introduced fuzzy set theory. It addresses vagueness and incomplete data used in various fields of science. Atanassov [2] addressed the short comings in fuzzy set theory with cultured way and opened a door to a new idea with new title intuitionistic fuzzy set theory. Molodtsov [3] inaugurated the concept of soft set theory to address the uncertainty. Soft set theory hugged with many applications in many fields, like smoothness of function, Riemann integration, measurement theory, and game theory. Molodtsov [4] continuously work on soft set theory. Alshehri et al. [5] applied the concept of soft set theory to K-algebra and traced some characteristics of Abelian soft K-algebras. In addition, they introduced the notion of intersection K-algebras and addressed some of their properties. Maji et al. [6] initiated first practical application of soft sets in decision-making problems. Feng et al. [7] for the first time considered the combination of soft sets, fuzzy sets, and rough sets. Using soft sets as the granulation structures, Feng et al. [8] defined soft approximation spaces, soft rough approximations, and soft rough sets, which are generalizations of Pawlak’s rough set model based on soft sets. It has been proved that, in some cases, Feng’s soft rough set model could provide better approximations than classical rough sets.

Xu et al. [9] unveiled the concept of vague soft set theory which is an extension to soft set theory. Huang et al. [10] deeply studied [9] and pointed out some incorrect results. They verified the incorrect results with examples and gave some more new definitions. Chang Wang and Yaya Li [11] initiated the concept of vague soft topological structures with title topological structure of vague soft sets. The authors discussed the basic concepts related to vague soft topological space and studied the results in vague soft topology. The soft set to the hyper-soft set was generalized by Smarandache [12]. In addition to this, the author introduced the hybrids of crisp, fuzzy, intuitionistic fuzzy, neutrosophic, and plithogenic hyper-soft set. Bera and Mahapatra [13] introduced the concept of neutrosophic soft topology and discussed some fundamental results. Ozturk in [14] introduced new concepts in neutrosophic soft topological spaces. These concepts are boundary, dense set, and neutrosophic soft basis. In addition, the concept of soft subspace on neutrosophic soft topological spaces. Some interesting results are addressed with respect to soft points. Some complicated results are secured with best examples.

Yolcu et al. redefined neutrosophic soft mapping and studied the images and inverse images of neutrosophic soft sets. The authors continued to trace the basic operations and other related properties of neutrosophic soft mapping. The authors beautifully applied neutrosophic soft mapping to application in decision-making problems.

Ozturk et al. [14] are pioneers of new operations on neutrosophic soft sets. Ozturk et al. [15] unveiled the concept of neutrosophic soft mapping, neutrosophic soft open mapping, and neutrosophic soft homeomorphism on the basis of operation defined in [14]. Some results are secured with best understandable examples. Gunduz et al. [16] introduced new concepts of neutrosophic soft sets. They defined new separation axioms in neutrosophic soft topological spaces with respect to soft points. In continuation, the relationship among these neutrosophic soft axioms has been addressed. Interior and closer of neutrosophic soft sets are also addressed. On the basis of these concepts, some other structures are also discussed. Most of the difficult results are secured with best examples.

AL-Nafee [17] introduced new family of neutrosophic soft sets. The author defined new operations on the neutrosophic set. These operations are union and intersection. On the basis of these new operations, the author defined neutrosophic soft topological space. AL-Nafee et al. [18] continued their work and extended the neutrosophic soft topological space to neutrosophic soft bitopological space on the basis of operations defined in [18]. The authors regenerated all the fundamental results of NSBTS on these basic operations. Dadas and Demiralp [19] inaugurated NSBTS on the basis of the operations defined in [13]. The authors introduced pairwise neutrosophic soft (closed) sets in NSBTS. These references [13–19] became source of motivation for my new research.

In our study, we have worked with the operations given in references [14–16] which are entirely different from references [13, 17]. In Section 2, some basic recipes are inaugurated. Originality begins in Section 3. In this section, new concepts of neutrosophic soft quad-topological spaces are addressed with examples. Some results, union and intersection are also studied in neutrosophic soft quad-topological spaces. In Section 4, some important definitions of generalized neutrosophic soft open sets in neutrosophic soft quad topological spaces are introduced. These definitions are semiopen, preopen, and open sets, respectively. These definitions became source of generation of different neutrosophical soft separation axioms and neutrosophical soft other separation axioms in neutrosophic soft quad topological spaces with respect to soft points of the second space. Neutrosophical soft quad homeomorphism which is a safe carriage for different structure from one space to another is defined. Soft closer attachment with neutrosophical soft separation axioms and neutrosophical soft other separation axioms in neutrosophic soft quad topological spaces with respect to soft points of the second space are addressed. In Section 5, more main results are addressed. Among neutrosophical soft separation axioms, Hausdorff space is considered to be the most important separation axioms. Important things should be given serious attention. So, this section has almost been engaged with the study of Hausdorff space. Through neutrosophical soft function, the characteristics of one space can be migrated to another space if the neutrosophical soft function is satisfying conditions of neutrosophical soft bijections and bicontinuousness. A soft function satisfying these conditions is known as neutrosophical soft homeomorphism. Neutrosophical soft homeomorphism is giving birth to neutrosophical soft topological property. Some neutrosophical soft topological properties of Hausdorff space are addressed with respect to soft points. Sequentially, compactness and countably compact in neutrosophic soft quad topological spaces with respect to soft points of the second space are addressed. Engagement of Hausdorff space with closed sets in neutrosophic soft quad topological spaces is addressed. In Section 6, some concluding remarks and future work are given.

2. Fundamental Concepts

In this section, fundamental concepts are addressed. These fundamental concepts are neutrosophic set, soft set, neutrosophic soft set, neutrosophic soft complement, neutrosophic soft subsets, neutrosophic soft union, neutrosophic soft intersection, null neutrosophic soft, neutrosophic soft absolute set, neutrosophic soft points, and neutrosophic soft topological spaces.

Definition 1. (see [20]). A neutrosophic set (NS) symbolized by A on the key set is defined as: , where

Definition 2. (see [3]). Let be the key set, be a set of parameters, and symbolizes the power set of . A pair is referred to as a soft set (SS) over where is a map given by .

Definition 3. (see [21, 22]). Let be the key set and set of parameters. Let signifies power set of all neutrosophic sets on . Then, a neutrosophic soft set over is a set defined by a set valued function representing a mapping , where is called approximate function of the neutrosophic soft set It can be written as a set of ordered pairs: .

Definition 4. (see [13]). Let be a neutrosophic soft set over key set the complement of is signified and is defined as follows:
It is clear that

Definition 5. (see [23, 24]). Let and are two neutrosophic soft set over key set . Then, is said to be neutrosophic soft subset of if , , , , and It is denoted by Neutrosophic soft is said to be neutrosophic soft equal to if is neutrosophic soft subset of and neutrosophic soft subset of It is dented by

Definition 6. (see [16]). Let be two neutrosophic soft subsets over key set so that , then their union is denoted by and is defined as where

Definition 7. (see [16]). Let and be two neutrosophic soft subsets over key set such that , then their intersection is defined as is given as where

Definition 8. (see [14]). Let be a neutrosophic soft set over a key set , then it is said to be a null neutrosophic soft set if
It is signified as

Definition 9. (see [14]). Let be a neutrosophic soft set over a key set , then it is said to be an absolute soft set if and for all
It is signified as ; clearly, .

Definition 10. (see [14]). Let NSS be the family of all neutrosophic soft sets and , then is said to be a neutrosophic soft topology on if : the union of any number of neutrosophic soft sets in : the intersection of a finite number of neutrosophic soft sets in in ; then, ) is said to be an over

Definition 11. (see [14]). Let be the family of all neutrosophic sets over , then neutrosophic set is neutrosophic point, for and is defined as follows:

Definition 12. (see [14]). Let be the family of all neutrosophic soft sets over KS Then, neutrosophic soft set is called a neutrosophic soft point if for every and is defined as follows:

Definition 13. (see [14]). Let (be a neutrosophic soft sets over KS , then ) if

Definition 14. (see [14]). Let be a soft neutrosophic topological space over Let be a neutrosophic soft set. Then, is called a neutrosophic soft neighborhood of the neutrosophic soft point if there exists a neutrosophic soft open set such that

3. Characterisation of Neutrosophic Soft Quad-Topological Structures

In this section, the concept of neutrosophic soft quad topological space is defined. Furthermore, new types of open and closed sets have been introduced in neutrosophic soft quad topological spaces.

3.1. Definition

If are four neutrosophic soft topological space (NSTS), then is called neutrosophic soft quad topological space. A neutrosophic soft subset is said to be neutrosophic soft quad open in if there exists a neutrosophic soft open set in neutrosophic soft open set in , neutrosophic soft open set in and neutrosophic soft open in such that

3.2. Example

Let , and , , , and where , and being as follows:

Therefore, , are on and so is a NSQTS.

3.3. Theorem

Let be a NSQTS. Then, is a on

Proof. For this, we have to verify all the three conditions of neutrosophic soft quad topological space. Conditions (1) and (3) are obvious; for condition (2), let , then , , , and , , and are NSTS on , then , , and . Therefore, .

Remark 1. Let be a NSQTS, then need not be a NSQTS on

3.4. Example

Let , and , , , and , where , and being NSSs as follows:

Here, is not a on.

3.5. Definition

Let be a NSQTS. Then, an is called as a pairwise NSQ open set if there exists a NSOS in in in and in such that for all such that

3.6. Definition

Let be a NSQTS. Then, an NSSis called as a PNSOS if there exist a NSOS in and a in in , and